Sketching Graphs Polynomial and Rational Functions

Sketching GraphsPolynomial and Rational FunctionsThe graphs shown here illustrate the properties of odd occurring roots versus even occurring roots, or more precisely when a specific root repeats an odd number of times versus an even number of times.The graph at the left is the graph of while the graph at the right is the graph of . Notice that the graph of passes through the x-axis changing from negative y-values to zero to positive y-values. The graph of however does not pass through the x-axis. It is simply tangent to the x-axis changing from positive y-values to zero and back to positive y-values. The same is true for each of the following examples. at the left and at the right. The only difference is a greater flattening of both functions at the zero and a greater increase (or decrease) in each of them as The same is true for each of the following examples. at the left and at the right. The only difference is even greater flattening of both functions at the zero and an even greater increase (or decrease) in each of them as The same is true for the graphs of at the left and at the right.A similar case can be made for restrictions, where there are sign changes in the value of the function on either side of a vertical asymptote determined by an odd occurring restriction and no sign change in the value of a function on either side of an even occurring restriction. As the degree of the function gets larger the rate of increase in the function as